Hello to everyone,
Could somebody please tell me how do we solve exercises like the one I describe below:
= e + O(1/n)
Thanks a lot,
Tasos
First of all, I want to thank you mmmbot for answering.
Sorry if I didn't write it well. By writing exp I meant e to the power.
So I mean: [e to the power of (1 + (O(1/n))^2)] = e + O(1/n)
P.S.1: Could you tell me how to write it in the post so as it will appear correctly?
P.S.2: What you say is right: exp(1) and e are the same thing. That is what I mean.
I'm checking to see whether or not O(1/n) is function notation ... be right back.
A big "Oh, please excuse me".
I thought O and n were constants. (I just saw the big Oh function for the first time at Wikipedia.)
No, you are correct. O(1/n) is good.
I cannot help you with this exercise.
Now, I read it as follows.
e^[1 + O(1/n)^2] = e + O(1/n)
Again, sorry for goofing up your thread.
~ Mark
Are you sure you put the parentheses at the right place? Because this is correct, but not optimal, as you shall see:
I'll apply usual properties of the "big O" notation. If you don't know them, just ask, the proofs are very short.
First you have , and it is more usual to write it this way.
Then .
I used the following expansion of the exponential at 0: when tends to 0, composed with the sequence which tends to 0.
Now, , hence the previous big O can be replaced by , and this is what you need.
Let us now suppose that the question was , which seems more plausible to me.
Then we would have: (as before, the second big O can be included in the first one), and by the same computation as above we conclude .
ps: about the way to write symbols, there's a section in the forum related to "LaTeX", and this is where you should look for help about this.
This is because of this result:
In fact, as soon as a function is differentiable at , we have as tends to 0 (because so that as ).
And if as , then , so that we can compose: . This can be written as tends to 0.
Now, in this expansion, you can replace by any sequence which converges to 0. For instance, . If , you get what I wrote and used.
(I'm thinking of something that may have been confusing: when I wrote , it meant , not exponential of the parenthesis)